Proving that a Set is Open: The ε-δ Criterion in Analysis

Understanding the concept of open sets in mathematics is foundational for advanced analysis. In this article, we will explore the set A {x, y : 0 x 1, 2 y 3}, which is an open interval in the plane, and prove that it is an open set using the ε-δ criterion in analysis. This proof will demonstrate the rigorous nature of the mathematical approach and its practical application.

Understanding the Set A

The set A is defined as the collection of all points (x, y) in the Cartesian plane such that 0 x 1 and 2 y 3. This describes an open rectangle (a rectangle without its boundary) in the plane. The open nature of this set is what gives us the flexibility to use the ε-δ criterion to prove its open-ness.

The ε-δ Criterion for Open Sets

The ε-δ criterion is a fundamental concept in mathematical analysis that is used to define limits and continuity, as well as to prove that sets are open. The criterion states that for any point p within the set, there exists a positive number ε such that the open disc of radius ε centered at p is entirely contained within the set. In mathematical terms, this can be written as:

For all points p (x, y) in A, there exists an varepsilon > 0 such that the open ball B(p, varepsilon) is contained in the set A.

Proving the Set A is Open

To prove that the set A is open, we will pick an arbitrary point p (x, y) within the set A and find an appropriate varepsilon 0 such that the open disc B(p, varepsilon) is entirely contained within A.

Step 1: Choosing the Point P

Consider an arbitrary point p (x, y) within the set A. This means that 0 x 1 and 2 y 3.

Step 2: Calculating the Distance to the Boundaries

The set A is bounded by the lines x 0, x 1, y 2, and y 3. The distance from the point p to these boundaries determines the size of the open ball we need. Specifically, the distances are:

The distance to the line x 0 is x. The distance to the line x 1 is 1 - x. The distance to the line y 2 is y - 2. The distance to the line y 3 is 3 - y.

We need to find the minimum of these distances, as this minimum distance will determine the radius varepsilon of the open ball centered at p.

Define:

varepsilon min{x, 1 - x, y - 2, 3 - y}

Since 0 x 1 and 2 y 3, it follows that:

x 0 1 - x 0 y - 2 0 3 - y 0

Therefore, varepsilon 0.

Step 3: Verifying the Open Ball is Contained in A

Consider the open ball B(p, varepsilon) centered at p (x, y) with radius varepsilon. The condition for a point q (u, v) to be in B(p, varepsilon) is that the Euclidean distance sqrt{(u - x)^2 (v - y)^2} is less than varepsilon.

For any point q (u, v) in B(p, varepsilon), we have:

[sqrt{(u - x)^2 (v - y)^2} varepsilon]

Since varepsilon min{x, 1 - x, y - 2, 3 - y}, each of these boundary distances guarantees that:

u x - varepsilon geq 0 u x varepsilon leq 1 v y - varepsilon geq 2 v y varepsilon leq 3

Therefore, q (u, v) must be within the bounds of A, proving that B(p, varepsilon) subseteq A.

Conclusion

In conclusion, we have shown that for any arbitrary point p (x, y) in the set A, we can find a positive ε such that the open ball centered at p with radius varepsilon is entirely contained within A. This fulfills the ε-δ criterion and proves that the set A is indeed an open set in the plane.

Understanding the ε-δ criterion and being able to apply it to prove the open-ness of sets is a crucial skill in mathematical analysis. This technique provides a rigorous foundation for more advanced topics in mathematics.